Lecture Notes Five

Fall 2001

The Normal Curve and Standard Scores

1. Minute Papers Past.

Of all the answers to the minute papers of last week I have selected some for mention here.

Here is the question again.

The deVoe Report (June 2, 1980) quoted then U.S. President Jimmy Carter as saying "half the people in this country are living below the median income -- and this is intolerable." What is disputable and what is true in this quote?
Here are some of the correct answers:
"President Carter must have meant that whatever the actual value of the median income was, it was intolerable that half of the nation lived below that income."
Indeed, that was the answer I was looking for.

The Carter quote was obviously taken out of context. We don't know what he said right before the quoted text, but he might have actually expressed the median income in dollars. Here's an extremely contrived version of this hypothesis, for illustration purposes:

"We have calculated the median yearly income and we found it to be (say) $600. We think this is a problem that needs to be addressed immediately; half the people in this country are living below the median income -- and this is intolerable."
There were some answers saying that Carter probably meant the mean. He did not mention the (arithmetical) mean, and that's probably because he did not want to say anything about it. He only wanted to make a comment about the median. He found it too low, and he expressed a concern that the value is too low to be the upper limit of income for half of the nation.

What makes this example intriguing is that, taken out of its context, it drastically polarizes our assumptions about what is said, placing the focus of our understanding on the wrong aspect. To see how this happens in another example (for your enjoyment) witness the following English sentence.

The ship sailed past the harbor sank.
How does this sound? Well, here's the same sentence in its original context:
A small part of Napoleon's fleet tried to run the English blockade at the entrance to the harbor. Two ships, a sloop and a frigate, ran straight for the harbor while a third ship tried to sail past the harbor in order to draw enemy fire. The ship sailed past the harbor sank.
I hope, perhaps, this makes the point. The question, and the quote, were tricky.

You need to watch for tricks like this in real life too.

Now the answers presented to the question posed last time follow. (So to speak).

Here is the answer in graphical form and some of the arguments:

In class we discussed the picture on the left. ("Roll the tape, Al!").

Says one student concisely, with an eloquent discourse:

  1. The mode is at the peak of the graph.

  2. The median will be to the left of the mode.

  3. The mean will be to the left of the mode as well. There are more scores on the left of the mode, which will lower the mean (compared to the mode).

  4. The median will be between the mean and the mode because it is only affected by number of values added. In this distribution we have the highest concentration of values around the mode. A lot of values need to be added for the median to even budge, given the height of the frequency distribution around the mode.

  5. Mean will be more affected because it takes into account all the values of the numbers added.
Let's now move to the topic for today, which is

2. The Normal Distribution

For many variables, most observations are concentrated near the middle of the distribution. As distance from the central concentration increases, the frequency of observation decreases. Such distributions are called "bell-shaped". An example is the normal distribution.

A broad range of observed phenomena in nature and in society is approximately normally distributed. For example, the distributions of variables such as

are approximately normal. The normal distribution is by far the most important theoretical distribution in statistics and serves as a reference point for describing the form of many distributions of sample data. Much of its importance occurs in conjunction with inferential statistics.

Take a look at the homework that is due tomorrow now.

3. Shape of the Normal Distribution

This is only an approximate rendering, still meaningful.

4. Area Contained Under The Normal Curve

Appears indicated in the diagram above.

Please check these numbers in lab.

5. Standard Scores (z-Scores)

A z score is a transformed score that designates how many standard deviation units the corresponding raw score is above or below the mean.

This transformation results in a distribution having a mean of 0 (zero) and a standard deviation of 1 (one). Again, we will need to verify this in the lab.

Important use of z scores:

to compare scores that are not otherwise directly comparable

Take a look at and think about the new homework again now.

6. Characteristics of z-Scores

  1. The z-scores have the same shape as the set of raw scores. (Transforming the raw scores into their corresponding z scores does not change the shape of the distribution. Not do the scores change their relative positions. All that is changed are the score values).

  2. The mean of the z scores always equals zero. This follows from the observation that the scores located at the mean of the raw scores will also be at the mean of the z scores.

7. Finding Areas Corresponding To Any Raw Score

In lab we will also look at the following transformations:

You'll see examples of these in lecture today.

Last updated: November 3, 2001 by Adrian German for A113